Italian, 2-rainbow and Roman domination numbers in middle graphs

Given a graph $G$, we consider the Italian domination number $\gamma_I(G)$, the $2$-rainbow domination number $\gamma_{r2}(G)$ and the Roman domination number $\gamma_R(G)$. It is known that $\gamma_I(G) \leq \gamma_{r2}(G) \leq \gamma_R(G)$ holds for any graph $G$. In this paper, we prove that $\gamma_I(M(G)) =\gamma_{r2}(M(G)) =\gamma_R(M(G)) =n$ for the middle graph $M(G)$ of a graph $G$ of order $n$, which gives an answer for an open problem posed by Mustapha Chellali et al. [Discrete Applied Mathematics 204 (2016) 22--28]. Moreover, we give a complete characterization of Roman domination stable middle graphs, 2-rainbow domination stable middle graphs and Italian domination stable middle graphs.